Dynamic Safety in Complex Environments: Synthesizing Safety Filters with Poisson's Equation

TL;DR

This paper introduces a novel method for generating safety sets for robotic systems in dynamic environments by solving Poisson's equation, enabling real-time safety filtering demonstrated on robots navigating obstacle-rich settings.

Contribution

It presents a new algorithm that synthesizes safety functions from perception data using elliptic PDEs, specifically Poisson's equation, for improved safety guarantees in robotics.

Findings

Successfully generates safety functions from perception data.

Enables real-time safety filtering on robots.

Demonstrates effectiveness in dynamic obstacle environments.

Abstract

Synthesizing safe sets for robotic systems operating in complex and dynamically changing environments is a challenging problem. Solving this problem can enable the construction of safety filters that guarantee safe control actions -- most notably by employing Control Barrier Functions (CBFs). This paper presents an algorithm for generating safe sets from perception data by leveraging elliptic partial differential equations, specifically Poisson's equation. Given a local occupancy map, we solve Poisson's equation subject to Dirichlet boundary conditions, with a novel forcing function. Specifically, we design a smooth guidance vector field, which encodes gradient information required for safety. The result is a variational problem for which the unique minimizer -- a safety function -- characterizes the safe set. After establishing our theoretical result, we illustrate how safety functions can be used in CBF-based safety filtering. The real-time utility of our synthesis method is highlighted through hardware demonstrations on quadruped and humanoid robots navigating dynamically changing obstacle-filled environments.